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Number · Standard form

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Calculating in standard form

Why multiplying and dividing in standard form is just the index laws, why the answer often needs re-standardising, and why adding needs the powers aligned first.

Number · Standard form

Calculating in standard form

Why multiplying and dividing in standard form is just the index laws, why the answer often needs re-standardising, and why adding needs the powers aligned first.

Why it works

Standard-form arithmetic is the index laws wearing a lab coat. Multiplication can be regrouped freely, so pair up the parts:

(3×104)×(2×105)=(3×2)×(104×105)=6×109.(3 \times 10^4) \times (2 \times 10^5) = (3 \times 2) \times (10^4 \times 10^5) = 6 \times 10^9.

Coefficients multiply; powers of ten add (never multiply — 102010^{20} is the index-law error in disguise). Division the same way: coefficients divide, powers subtract —

8×1073.2×102=83.2×107−2=2.5×105.\frac{8 \times 10^7}{3.2 \times 10^2} = \frac{8}{3.2} \times 10^{7-2} = 2.5 \times 10^5.

The answer must be re-standardised. The arithmetic is oblivious to the 1≤a<101 \le a < 10 rule, so it often lands outside it:

(4×105)×(5×103)=20×108.(4 \times 10^5) \times (5 \times 10^3) = 20 \times 10^8.

That's the right value in the wrong form. Trade: 20=2×1020 = 2 \times 10, so 20×108=2×10920 \times 10^8 = 2 \times 10^9. The trade always balances — coefficient ÷10 ⇒ power +1; coefficient ×10 ⇒ power −1 (so 0.4×106=4×1050.4 \times 10^6 = 4 \times 10^5, the power goes down). Sense-check the direction: the value must not change.

Adding and subtracting: align the powers first. Powers of ten are place value, and you can only add digits sitting in the same places:

3.4×105+2.7×104=3.4×105+0.27×105=3.67×105.3.4 \times 10^5 + 2.7 \times 10^4 = 3.4 \times 10^5 + 0.27 \times 10^5 = 3.67 \times 10^5.

Adding the powers (10910^9) or just adding coefficients across different powers are both meaningless — same reason 340+27340 + 27 isn't 6161.

Negative powers change nothing. The laws carry through: 6×1081.5×10−3=4×108−(−3)=4×1011\frac{6 \times 10^8}{1.5 \times 10^{-3}} = 4 \times 10^{8-(-3)} = 4 \times 10^{11} — subtracting a negative power pushes the answer up, because dividing by a tiny number gives a huge one. Use that as the sanity check.

Calculator displays are not answers. A screen showing 3.2E11 or 3.2113.2^{11} means 3.2×10113.2 \times 10^{11} — copy it as standard form, never as "3.2 to the power 11".